Question

Write down the conjugate of each of the following:

(i)$(-5+\sqrt{-1})$

(ii)$(-6-\sqrt{3})$

(iii)$i^3$

(iv)$(4+5i{)}^2$

Answer 1

(i) Let $z=-5+\sqrt{-1}=(-5+i)$. Then,
$\overline{z}=\overline{(-5+i)}=(-5-i).$

(ii) Let $z=(-6-\sqrt{3})=-6-i\sqrt{3}$. Then,
$\overline{z}=\overline{(-6-i\sqrt{3})}=(-6+i\sqrt{3})$

(iii) Let $z=i^3=-i=0-i$. Then,
$\overline{z}=\overline{(0-i)}=(0+i)=i$.

(iv) Let $z=(4+5i{)}^2=(4{)}^2+(5i{)}^2+2\times 4\times 5i$
$=(16-25+40i)=(-9+40i).$
$\therefore \overline{z}=\overline{(-9+40i)}=(-9-40i)$